Two moles of an ideal monatomic gas occupies a volume V at 27 °C. The gas expands adiabatically to a volume 2V. Calculate (i) the final temperature of the gas and (ii) change in its internal energy.

Options

A. (a) 189 K (b) -2.7 kJ
B. (a) 195 k (b) -2.7 kJ
C. (a) 189 K (b) 2.7 kJ
D. (a) 195 K (b) -57 kJ

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the initial conditions
We have:
- Number of moles (n) = 2 moles
- Initial volume (V_i) = V
- Final volume (V_f) = 2V
- Initial temperature (T_i) = 27 °C = 300 K (since we convert Celsius to Kelvin by adding 273)

### Step 2: Determine the value of gamma (γ)
For a monatomic ideal gas, the value of gamma (γ) is given by:
\[ \gamma = \frac{C_p}{C_v} = \frac{5}{3} \]

### Step 3: Use the adiabatic process relation
For an adiabatic process, the relationship between temperature and volume is given by:
\[ T_i V_i^{\gamma - 1} = T_f V_f^{\gamma - 1} \]

Substituting the known values:
\[ T_i V_i^{\frac{2}{3}} = T_f (2V)^{\frac{2}{3}} \]

### Step 4: Rearranging the equation
We can express the final temperature (T_f) in terms of the initial temperature (T_i):
\[ T_f = T_i \left( \frac{V_i}{V_f} \right)^{\gamma - 1} \]

Substituting the values:
\[ T_f = 300 \left( \frac{V}{2V} \right)^{\frac{2}{3}} \]
\[ T_f = 300 \left( \frac{1}{2} \right)^{\frac{2}{3}} \]

### Step 5: Calculate the final temperature
Now, calculate the value:
\[ T_f = 300 \times \left( \frac{1}{2} \right)^{\frac{2}{3}} \]
\[ T_f = 300 \times \frac{1}{\sqrt[3]{4}} \]
\[ T_f \approx 300 \times 0.7937 \]
\[ T_f \approx 238.11 \, \text{K} \]

### Step 6: Calculate the change in internal energy (ΔU)
The change in internal energy for an ideal gas is given by:
\[ \Delta U = \frac{f}{2} n R \Delta T \]
Where:
- \( f \) for a monatomic gas = 3
- \( R \) = 8.31 J/(mol·K)
- \( \Delta T = T_f - T_i = 238.11 - 300 = -61.89 \, \text{K} \)

Now substituting the values:
\[ \Delta U = \frac{3}{2} \times 2 \times 8.31 \times (-61.89) \]
\[ \Delta U = 3 \times 8.31 \times (-61.89) \]
\[ \Delta U \approx -1547.51 \, \text{J} \]

### Step 7: Convert to kilojoules
\[ \Delta U \approx -1.55 \, \text{kJ} \]

### Final Answers:
(i) The final temperature of the gas is approximately **238.11 K**.
(ii) The change in internal energy is approximately **-1.55 kJ**.